To investigate if early warning signals are vulnerable to this fallacy,
we simulate a system that is not driven towards a bifurcation such as
in Fig reffig:1(b). This simulation approach allows us to determine whether
examining historical events is a valid way to test the utility of these
indicators. We simulated 20,000 replicates of a stochastic individual-based
birth-death process with an Allee threshold [1]}, which
arises from positive fitness effects at low densities. Above the Allee
threshold the population returns to a positive equilibrium size, whereas
below the threshold the population decreases to zero. The model can be represented as a continuous time birth-death process where births and deaths are Poisson events which depend on the current density with rates given by
\(b(n) &= \frac{ K n^2}{n^2 + h^2},  \\d(n) &= e n + a,\) 
